The second moment of a certain pair correlation function for Sato-Tate sequences

Abstract

In [BS19], Balasubramanyam and Sinha derived the first moment of the pair correlation function for Hecke angles lying in small subintervals of [0, 1], as one averages over large families of Hecke newforms of weight k with respect to Γ0(N). The goal of this thesis is to study the second moment of this pair correlation function. We also record estimates for lower order error terms in the computation of the second moment and show that the variance goes to 0 under the same growth conditions on weights and levels for the families of Hecke newforms as required for the convergence of the first moment.